MATH2070/2970: Optimisation and Financial Mathematics Semester 2, 2022
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Computer Project
MATH2070/2970: Optimisation and Financial Mathematics
Semester 2, 2022
The Scenario: Consider a universe with five stocks:
Stock: P1 P2 P3 P4 P5
Code: BHP CSL NAB REA WOW
Anthony has a risk aversion parameter t = 0.2. He wants to invest $200,000 in the five stocks.
1. Import the dataset UFM projdat c-t-．asv which contains the adjusted daily closing prices from 1/10/2021 to 30/9/2022 of the five stocks.
(a) For each of the five stocks, compute the simple return rates, that is, Rt = StSt(一)
, where
St is the price of the stock at time t. Estimate the mean vector and covariance matrix of the returns, round to 4 decimal places, and verify that they are of the following:
┌ 
┐
r = '(') 0.0005 '(') ,
''
┌0.00(0)04
' 0
0
0.0002
0
0.0002
0.0001
0
0 0.0002 0.0001 0.0001
0 0.0002 0.0001 0.0006 0.0001
0.00(0)01┐
0.0001'(')
0.0001 '
(Use these rounded µ and S for subsequent analysis.)
(b) Determine which risk-averse investors (in terms of the values of t) short sell in this market
and which funds they short sell. Are there any funds that no-one will short sell or that everyone will short sell?
2. Anthony’s Optimal Portfolio: Carry out the following computational tasks for Anthony’s optimal portfolio P* .
(a) Obtain Anthony’s optimal proportion and dollar investment in each of the five funds. Give
also the expected return (µ) and risk (σ) of P* .
(b) Suppose Anthony doesn’t want to short sell any asset. Obtain the optimal restricted
portfolio P for Anthony.
(c) Illustrate the problem graphically by plotting the following (on the same graph) in the µσ-plane:
(i) The five stocks.
(ii) The Minimum Variance Frontier. Use a t-range |t| ≤ 0.3 for your display. (iii) The minimum risk portfolio.
(iv) 1000 random feasible portfolios satisfying |xi| ≤ 1 (for each of the 5 funds) and σi ≤
0.03 for i = 1, . . . , 1000.
(v) Anthony’s indifference curve, unrestricted portfolio P* and restricted portfolio P .
3. Adding a Riskless Cash Fund: Suppose now that a riskless cash fund P0 is also available to invest in. The risk free rate is 0.0001 for both lending and borrowing.
(a) Obtain the new optimal (unrestricted) portfolio 
allocation of Anthony to the (now) six
(b) Give the efficient frontier (Capital Market Line) of the six assets and the tangency portfolio.
(c) Make a new µσ-plane graph showing the riskless cash fund P0, the tangency portfolio,
Anthony’s new optimal portfolio 
and the Capital Market Line, in comparison to the
4. Capital Asset Pricing Model: Take the tangency portfolio obtained in Q3(b) as the market portfolio PM .
(a) Compute the β’s of all relevant assets in this project (5 stocks, P0 , P* , P , 
, PM).
return r6 = 0.0005 and a covariance with the market portfolio Cov(R6, RM ) = 0.0003. According to CAPM, what should the expected return of the asset be? Compare this with the given estimated expected return.
(c) Produce a plot in µβ-plane showing the Security Market Line and the β’s of all relevant assets in this project (5 stocks, P6 , P0 , P* , P , 
, PM).
5. (Advanced) There are different ways in analysing the return rates. Instead of the simple return rates, one can also look at the log return rate, that is, Yt = ln St − ln St一1 = ln 
, where St is the price of the stock at time t.
(a) From the dataset, estimate the mean vector and covariance matrix of the log return rates
of the five stocks.
(b) For each of the five stocks, construct a histogram of the log return rates and superimpose
a Normal density curve (with parameters estimated in part (a)) on the histogram. What can you say from your plots?
(c) Suppose that Yt has a Normal distribution N(µ, σ2 ). Let Qt = 
. Compute (alge- braically) the expected value and variance of Qt (in terms of µ and σ).
2022-10-18